The proof of Theorem REF  shows that for fixed values of \(\theta \)  and \(\rho \) , the Lyapunov \(\lbrace \Phi ^k\rbrace \)  must be monotonically decreasing, whenever the proximal weights \(\tau _x\)  are sufficiently large.
To verify this empirically, we fix \(\theta = 10^6\) , \(\rho = 1\) , \(\tau _z = \kappa _z \rho \)  and then consider \(\tau _x \in \lbrace 0, 1, 2\rbrace \) ; we also disable their automatic updates in Algorithm REF .
Figure REF  shows the values of the corresponding Lyapunov sequences \(\lbrace \big (\Phi ^k - \underline{\Phi }\big )/{\underline{\Phi }}\rbrace \) , where \(\underline{\Phi }\)  is a normalizing constant equal to the smallest observed value of \(\Phi ^k\)  (across 100 iterations).
We find that when the proximal terms are absent (\(\tau _x = 0\) ) or when they are present but not sufficiently large (\(\tau _x = 1\) ), the Lyapunov sequence (and hence also \(\lbrace (x^k, z^k)\rbrace \) ) diverges.
This is contrast to classical Gauss-Seidel based algorithms [1]}, [2]}, which are not as critically dependent on proximal terms.
Indeed, the Jacobi nature of the \(x\) -update in Algorithm  requires that the new iterate \(x^k\)  be sufficiently close to its old value \(x^{k-1}\)  for convergence.
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